The Mechanics of Rocking Stones: Equilibria on Separated Scales

Rocking stones, balanced in counter-intuitive positions, have always intrigued geologists. In our paper, we explain this phenomenon based on high-precision scans of pebbles which exhibit similar behavior. We construct their convex hull and the heteroclinic graph carrying their equilibrium points. By systematic simplification of the arising Morse–Smale complex in a one-parameter process, we demonstrate that equilibria occur typically in highly localized groups (flocks), the number of the latter being reliably observed and determined by hand experiments. Both local and global (micro and macro) equilibria can be either stable or unstable. Most commonly, rocks and pebbles are balanced on stable local equilibria belonging to stable flocks. However, it is possible to balance a convex body on a stable local equilibrium belonging to an unstable flock and this is the intriguing mechanical scenario corresponding to rocking stones. Since outside observers can only reliably perceive flocks, the last described situation will appear counter-intuitive. A comparison between computer experiments and hand experiments reveals that the latter are consistent, that is, the flocks can be reliably counted and the pebble classification system proposed in our previous work is robustly applicable. We also find an interesting logarithmic relationship between the flatness of pebbles and the average number of global equilibrium points, indicating a close relationship between classical shape categories and the new classification system.